8.4: Nonradial Oscillations

So far we have considered only those oscillations that involve the radial coordinate only. While these oscillations seem sufficient to explain the majority of known pulsating stars, other less dramatic phenomena result from more complicated oscillations. Indeed, one would expect that most pulsational energy would appear in the fundamental radial mode of oscillation, and it is precisely those modes involving the modulation of the greatest amount of energy that can be most easily detected. However, the detection of short-period oscillations of low amplitude in the sun suggests that more complicated types of oscillations can occur. Their importance to the structural models of the sun and their probable detection in some early-type stars require that we spend a little time discussing them. However, the subject is too broad and many of the results are too uncertain to do more than sketch the nature of the problem. To give the greatest insight into the nature of the problem, I will concentrate on the adiabatic oscillations. The true cause of the oscillations lies in nonadiabatic theory, as it did for radial oscillations, and the results are still rather uncertain. In addition, the theory for oscillations among stars that are not spherically symmetric is still in its infancy. It was clear from Chapter 6 that the loss of spherical symmetry resulted in a substantial increase in the complexity of the theoretical description. No less is to be expected from pulsation theory. From the small amount of energy involved in the present cases of nonradial oscillations, it will be appropriate to use perturbation theory and to assume that the amplitudes of the oscillations are small.

a. Nature and Form of Oscillations

Just as there exists a wave equation for radial oscillations, so there is a wave equation for nonradial oscillations. However, instead of being a scalar equation in the radial coordinate r alone, it will be a vector equation whose solution will represent the behavior of the displacement vector \(\delta\mathrm{\vec{r}}\) and the associated variations of the state variables in the various dimensions that define the star. Since our problem will be an adiabatic one, the solutions will be undamped waves propagating not only along the radial coordinate but also over the surface coordinates. In Chapter 7 we saw that it was possible to represent the polar angle variation in terms of a series of Legendre polynomials. This was a special case of a much more general representation of the angular variation of solutions to a wide range of important physical equations. Laplace's equation, the Schrödinger equation, and the wave equation of classical electrodynamics are only a few of the equations whose solutions can be described in terms of spherical harmonics. Spherical harmonics are basically the product of the elements of two sets of functions. One set of functions describes the solution variation in the polar angle and is represented by the orthonormal Legendre polynomials. The orthogonal functions that describe the behavior in the azimuthal coordinate are just \(\mathrm{e^{i m \phi}}\). Thus the spherical harmonics are defined¹³ as \[Y_l^m(\theta, \phi)=(-1)^m\left[\frac{2 l+1}{4 \pi} \frac{(l-m)!}{(l+m)!}\right]^{1 / 2} P_l^m(\cos \theta) e^{i m \phi}\label{8.3.1}\]

As long as the star remains spherical, the equation describing the nonradial oscillations will also be separable in spherical coordinates, and the orthogonality of the spherical harmonics will guarantee that the full solution can be represented in terms of them. Since some of the solutions to the radial wave equation were standing waves, we should not be surprised if that were also the case for the nonradial oscillations. The linearity of the small-oscillation equations guarantees that the sum of any two solutions is also a solution. So a standing wave is just the interference pattern of two oppositely directed traveling waves of similar amplitude. Just as there could be higher-order modes of the radial wave equation excited, so higher-order waves in the azimuthal and polar angles can be present. However, different modes in each of the coordinates may combine to provide a much richer multiplicity of possible oscillations for the nonradial case. Thus the solution will have the form \[\xi(r, \theta, \phi, t)=\operatorname{Re}\left|\Re(r) P_l^m(\cos \theta) e^{i(m \phi+\omega t)}\right|\label{8.3.2}\]

The quantity \(\Re\) is a function of the radial coordinate only and represents the eigenfunctions that are possible in the radial direction. Different orders are usually denoted by the letter \(n\). The periodic behavior of the spherical harmonics implies that the parameters \(l\) and \(m\) will denote the different eigenfunctions in the polar and azimuthal coordinates, respectively.

Deubner and Gough¹³ give a nice way of seeing this. Consider cases when the wavelength of the oscillation is much less than the stellar dimensions. Under these conditions Lamb¹⁴ showed that the equations of motion can be reduced to the familiar form \[\frac{d^2 \psi}{d r^2}+K^2 \psi=0\label{8.3.3}\]

where \[\psi=\rho^{1 / 2} c_s^2 \nabla \cdot \vec{\xi}\label{8.3.4}\]

and \(\mathrm{c_s}\) is just the local speed of sound. This is simply the equation for a simple harmonic oscillator; here \(K\) is the local wave number and in this instance is related to the frequency of oscillation ω, scale height h, and local gravity \(g\) by \[K^2=\frac{\omega^2-\omega_c^2}{c_s^2}+\frac{l(l+1)}{r^2}\left(\frac{N^2}{\omega^2}-1\right)\label{8.3.5}\]

where \[\omega_c^2=\frac{c_s^2(1-2 d h / d r)}{4 h^2} \quad N^2=g(1 / h)-g / c_s^2\label{8.3.5a}\]

Thus we may expect that the general solution for the equations of motion will consist of a complicated interplay of waves propagating in all three coordinates. Also the specific nature of these waves will depend on the structure of the star, with the low-frequency wave anchored deep in the interior and the high-frequency waves determined largely by the local structure of the star nearer the surface. To try to bring some order to the multiplicity of oscillations that may be present in stars, let us consider an idealized case.

b. Homogeneous Model and Classification of Modes

Consider a homogeneous star of uniform density. Admittedly this is an unrealistic case in the real world, but it has the virtue that the eigenfrequencies of the equations of motion can be found and have a particularly simple form. Cowling¹⁵ found that the eigenfrequencies could be organized into several groups based on the physical mechanisms primarily responsible for their propagation. These modes all have their counterparts in the solutions of more realistic models, and so Cowling's classification scheme provides a useful basis for identifying the types of modes to be expected in real stars. Following Cox⁷ (p. 235), we can define a dimensionless frequency \[\omega_{l n}^2 \equiv \frac{\sigma_{l n}^2 R^3}{G M}\label{8.3.6}\]

The radial oscillation modes are then given by \[\begin{aligned}
& \omega_{l n}^2=Q_{l n} \pm\left[Q_{l n}^2+l(l+1)\right]^{1 / 2} \\
& Q_{l n} \equiv-2+\frac{\Gamma_1[n(2 l+2 n-5)+2 l+3]}{2}
\end{aligned}\label{8.3.7}\]

which for large \(n\) are approximately given by \[\omega_{l n}^2 \approx \begin{cases}2 \Gamma_1 n^2+\frac{l(l+1)}{2 \Gamma_1 n^2} & p \text { modes } \\ \frac{-l(l+1)}{2 \Gamma_1 n^2} & g \text { modes }\end{cases}\label{8.3.8}\]

The negative root in equation \ref{8.3.7} and its asymptotic counterpart in the g modes of equations \ref{8.3.8} imply that \(f^2_{\ln}<0\). So the star is dynamically unstable, and this is a result of the homogeneous model's being unstable to convection. In real stars this is not generally the case, and the g modes can be real.

The terminology has its roots in the nature of the oscillations corresponding to each of the modes. The \(p\) modes are known as pressure modes; they can be viewed as pressure or acoustic waves and are characterized by relatively large radial pressure disturbances. For n = 0 they correspond to the radial oscillations studied in the previous two sections. Thus, as n increases, the \(p\) modes can be roughly viewed as radial standing waves having n nodes. It would be reasonable to call them longitudinal waves. On the other hand, stable oscillations characterized by the \(g\) modes can be viewed as transverse waves. They are also known as gravity waves (not to be confused with gravitational waves, which are a phenomenon of the general theory of relativity); because the primary force acting as a restoring force for the oscillation is the local gravity. These waves are characterized by relatively small pressure and density variations and are largely transverse in their physical displacement. The most common analogy to these waves is water waves where the restoring force is clearly that of gravity and virtually no pressure or density changes are involved. Curiously the case for \(n = 1\) and \(n >> 1\) leads to \[\omega_{n n}^2 \approx-\frac{1}{4 \Gamma_1}\label{8.3.9}\]

and a characteristic frequency that is independent of the order \(n\). Physically such a condition would correspond to small blobs of material having a typical size much less than the stellar dimension, moving radially, and exhibiting small pressure and density changes. This is a fairly good description of a convective element and is often taken as a basis for describing the expected spectrum for convective blobs in a region unstable to convection. Thus the presence of g modes in a region stable against convection may be the result of excitation by a lower-lying convective region. The actual frequencies of oscillation for the g modes are always less than those for the corresponding p mode (see Figure 8.1).

Lying between these two classes of modes is a solitary mode called the \(f\) mode. This mode is generally attributed to Lord Kelvin and is characterized by \(\Delta\cdot\delta\mathrm{r}=0\) for the homogeneous model. This implies that both \(\delta P\) and \(\delta\rho\) are zero as for an incompressible fluid. However, this is not true for stellar models in general, and so this mode is sometimes referred to as the pseudo-Kelvin mode. Its dimensionless eigenfrequency for the homogeneous model is given by \[\omega_f^2=\frac{2 l(l-1)}{2 l+1}\label{8.3.10}\]

This eigenfrequency is independent of \(\Gamma_1\) which is to be expected since \(\delta P\) and \(\delta \mathrm{p}\) are both zero. That condition also implies a link between the radial and angular displacements so that oscillations in the radial coordinate are uniquely linked to displacements in the radial coordinate. There are no stable modes for \(l<2\).

Figure 8.1 shows the spectral distributions of modes to be expected for non-radial oscillations. The vertical axis indicates the angular eigenvalues while the horizontal axis displays the corresponding oscillation frequency. Therefore the pure radial modes lie on the horizontal axis (\(l=0\)) at the end of the spectra for \(p\) modes. The negative \(g\) modes are those for which the model is unstable and their frequencies indicate the growth of the instability. (reprinted from Cox,J.P. "Theory of Pulsation" pp239)

c. Toroidal Oscillations

There remains one last class of oscillations that we have not considered. So far we have been faithful to our assumption of spherical symmetry and discussed no modes that have a dependence on the azimuthal angle \(\phi\) so that \(m=0\). To have included cases where \(m \neq 0\) would have been to admit the existence of a preferred plane and thereby violate the assumption of spherical symmetry. Thus the modes described so far will present surface phenomena that are independent of the orientation of the star. This will not be the case for \(m \neq 0\). However, there is no a priori reason why azimuthal modes cannot exist. Indeed, for each value of l there are \(2l+1\) allowed values of \(m\) (that is, \(m=0\), \(\forall 1\), \(\forall 2\), \(\forall \cdots \forall 1\)) which represent waves that propagate in the \(\forall\phi\) direction. Of course for a nonrotating star there is no preferred direction of propagation, so these modes are degenerate and there are only \(l+1\) distinct possibilities.

Papaloizou and Pringle¹⁶ have shown that for rotation, this degeneracy is broken and the resulting modes correspond to traveling waves around the rotational axis of the star similar to Rossby waves in the earth's atmosphere, so they designated them r modes. These waves travel with a characteristic velocity that is approximately 1/m times the rotational period of the star. Thus such a wave would be seen by an observer to be moving at a rate that is slightly faster (\(+m\)) and slightly slower (\(-m\)) than the rotational speed of the star. Any comprehensive analysis of the effects of rotation must deal with the effects of angular momentum conservation as well as shape distortion and is therefore quite difficult. However, there can be little doubt that rotation will influence the values for the eigenfrequencies for the \(p\) and \(g\) modes. Although the overall effects of rotation are extremely complicated, there is some evidence from nonadiabatic studies that the prograde modes are somewhat less stable and therefore more likely to be excited, than the retrograde modes.

It is possible to have such modes in a star for which the total angular momentum is zero. In the case where \(l = 1\) such modes represent uniform rotation of the object. For \(l > 1\) the modes would represent torsional oscillations. In the simplest case where \(l = 2\), one hemisphere would rotate, say clockwise, while the other hemisphere rotated counterclockwise. Unfortunately, to stabilize such an oscillation, some sort of shear would have to be sustained by the stellar material. Such restoring shears are simply absent in normal stars. However, in white dwarfs and neutron stars, at least part of the star is expected to be in a crystalline phase and this matter could perhaps sustain such oscillations.

d. Nonradial Oscillations and Stellar Structure

From all that we have discussed so far, clearly the spectrum of oscillations present in a star depends critically on the structure of the star. If it were possible to observe the full spectrum of these oscillations, including their frequency and amplitude, quite sensitive tests of the internal structure of the equilibrium model would be possible. The analogy has been made to geophysicists who deduce the internal structure of the earth from the propagation of seismic waves produced by earthquakes. So strong is this analogy that the term helioseismology has come into fairly common use to describe oscillation analysis as applied to the sun.

Clearly, in the case of the sun, we have an opportunity to map the oscillation structure with a high degree of accuracy. While research in this area is ongoing, both \(p\) and \(g\) modes have been detected. The p modes are usually characterized by the generic term 5-minute oscillations. Oscillations described by \(1 < l < 1000\) have been detected, and the power (amplitude) of those oscillations has been determined. This is done by mapping the radial velocity field of the entire solar disk over extended periods and performing a Fourier analysis of the result for periodic structures. Clearly the velocity of an oscillatory displacement is simply related to the displacement and to the frequency itself. Thus the highly accurately determined velocity measurements provide accurate knowledge of the amplitude and frequencies of the oscillations that are present. The ability to resolve closely spaced frequencies simply requires a long, continuous series of observations.

In addition to the \(p\) modes, \(g\) modes with characteristic periods ranging from just under 3 hours to nearly 6 days have been reported. There is some disagreement among observers as to what constitutes actual periodic waves and which modes are "aliases" of other periodic phenomena and the data-sampling procedure. However, the existence of the \(g\) modes is quite likely. In general, the low-order modes represent wave structures that penetrate deeply into the interior, while the higher-frequency modes are confined to the outer layers of the sun. Indeed, the highest-frequency \(p\) modes are probably confined to the solar atmosphere itself. Thus the full spectrum of the solar oscillations allows a continuous depth probe of the internal solar structure. The standard solar model reproduces the overall properties of the oscillation spectrum, but fails to fit that spectrum in detail. There is some indication that the standard solar model may have a helium abundance that is somewhat low. Ulrich and Rhodes¹⁷ conclude that the failure of the standard solar model to fit the observations of the oscillation spectrum is real and lies outside the errors of either theory or observation. This would place it in the same category as the solar neutrino problem. Hopefully the solution of one can provide a solution for the other.

There are strong indications that nonradial modes have been detected in other stars. β Cephei stars are suspected to exhibit the effects of traveling waves on their surfaces in their spectra. Papaloizou and Pringle¹⁶ explain the short period oscillations seen in some cataclysmic variables to the r modes of surface traveling waves. In addition^{18, 19} sharp absorption features that move through the broad absorption lines of some rapidly rotating stars have been interpreted as representing nonradial oscillations. If this is proves to be the case, then the observations indicate the existence of a phenomenon for which there is no clear theoretical description. For reasons already mentioned, pulsation theory in the presence of extreme rotation is extremely difficult and far from well developed. However, should nonradial oscillations be unambiguously measured for these stars, the potential for detailed understanding of their internal structure is considerable. Given the uncertainties regarding the effects of rotation on the internal structure of these stars, every effort should be made to explore these observations as a probe of the stellar interior.